What is "central limit theorem"

alt. 1 (context statistics and mathematics singulare tantum English) The theorem that states that if the sum of independent identically distributed random variables has a finite variance, then it will be approximately normally distributed. 2 (context mathematics countable English) Any of various similar theorems. n. 1 (context statistics and mathematics singulare tantum English) The theorem that states that if the sum of independent identically distributed random variables has a finite variance, then it will be approximately normally distributed. 2 (context mathematics countable English) Any of various similar theorems.

In probability theory, the central limit theorem (CLT) states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined (finite) expected value and finite variance, will be approximately normally distributed, regardless of the underlying distribution. To illustrate what this means, suppose that a sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic average of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the computed values of the average will be distributed according to the normal distribution (commonly known as a "bell curve"). A simple example of this is that if one flips a coin many times, the probability of getting a given number of heads should follow a normal curve, with mean equal to half the total number of flips.

The central limit theorem has a number of variants. In its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations, given that they comply with certain conditions.

In more general usage, a central limit theorem is any of a set of weak-convergence theorems in probability theory. They all express the fact that a sum of many independent and identically distributed (i.i.d.) random variables, or alternatively, random variables with specific types of dependence, will tend to be distributed according to one of a small set of attractor distributions. When the variance of the i.i.d. variables is finite, the attractor distribution is the normal distribution. In contrast, the sum of a number of i.i.d. random variables with power law tail distributions decreasing as |x| where 0 < α < 2 (and therefore having infinite variance) will tend to an alpha- stable distribution with stability parameter (or index of stability) of α as the number of variables grows.